If they commute, then yes, each term in the series is just [tex]A^n B^n[/tex]. Otherwise you have to just calculate a LOT of commutators and hope something nice works out. What specific problem were you trying to tackle?

Where the operators in question are photon number operators in two different radiation modes.

I understand that there exists a decomposition (besides the taylor series expansion) of this operator using the Hubbard-Stratonovich decomposition. However, this decomposition requires that the creation and annihilation operators are fermomic. I, however, need a corresponding decomposition for bosonic operators.

The formula for exponential operator expansions of this sort is called the Baker-Campbell-Haussdorf formula which holds regradless if the operators are bosonic or fermionic and is a result of the mentioned taylor series expansion.

The formula for exponential operator expansions of this sort is called the Baker-Campbell-Haussdorf formula which holds regradless if the operators are bosonic or fermionic and is a result of the mentioned taylor series expansion.